Root
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Conference Proceedings
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4th International Conference on Physics and Control (PhysCon 2009)
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On nonlinear resonance oscillations of a spring supported point particle
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Full research of flat small nonlinear oscillations

of a spring pendulum with nonlinear dependence

of a tension of a spring on its lengthening is conducted.

The method of a hamiltonian normal form is used. For

reduction to a hamiltonian normal form the method of

invariant normalisation is used, what essentially reduces

calculations. Solutions of the normal form equations

have shown that periodic reorganisation between vertical

and horizontal oscillations occurs only in case of

resonances 1:1 and 2:1. At a resonance 2:1 this effect

is shown in square-law members of the equation, and

at a resonance 1:1 one should take into account cubic

members. In all other cases, both in the presence of

a resonance, and at its absence, oscillations have constant

frequencies with a little different from frequencies

of linear approach. For a resonance 2:1 it is found

maximum detuning of frequencies at which the effect

of swapping of energy from one kind of oscillation to

another disappears. The resonance 1:1 is physically

possible only for a spring possessing the negative cubic

term in the law of deformation.

of a spring pendulum with nonlinear dependence

of a tension of a spring on its lengthening is conducted.

The method of a hamiltonian normal form is used. For

reduction to a hamiltonian normal form the method of

invariant normalisation is used, what essentially reduces

calculations. Solutions of the normal form equations

have shown that periodic reorganisation between vertical

and horizontal oscillations occurs only in case of

resonances 1:1 and 2:1. At a resonance 2:1 this effect

is shown in square-law members of the equation, and

at a resonance 1:1 one should take into account cubic

members. In all other cases, both in the presence of

a resonance, and at its absence, oscillations have constant

frequencies with a little different from frequencies

of linear approach. For a resonance 2:1 it is found

maximum detuning of frequencies at which the effect

of swapping of energy from one kind of oscillation to

another disappears. The resonance 1:1 is physically

possible only for a spring possessing the negative cubic

term in the law of deformation.